topic 1: factors and multiples

Texas Grade 6: Scope and SequenceTable of Contents

Grade 6 Textbook: Table of Contents 105/25/21 *Pacing listed in 45-minute days

1 Composing and DecomposingTopic 1: Factors and MultiplesELPS: 1.A, 1.C, 1.E, 1.F, 1.G, 2.C, 2.E, 2.I, 3.D, 3.E, 4.B, 4.C, 5.B, 5.F, 5.G

Lesson Lesson Title Lesson Overview Essential Ideas TEKS Pacing*

1

Taking Apart Numbers and ShapesWriting Equivalent Expressions Using the Distributive Property

Students divide area models in different ways to see that the sum of the areas of the smaller regions equals the area of the whole model. They then rewrite the product of two factors as a factor times the sum of two or more terms, leading to the formalization of the Distributive Property.

• The area of a rectangle is the product of its length and width.

• An area model of a rectangle with side lengths a and (b + c) can be used to illustrate the Distributive Property.

• The Distributive Property states that for any numbers a, b, and c, a(b + c) = ab + ac.

• Equivalent expressions can be rewritten using properties.

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2

Searching for Common GroundIdentifying Common Factors and Common Multiples

Students construct rectangles with given areas and relate their dimensions to factors and common factors. They create prime factorizations to determine the greatest common factor (GCF) and least common multiple (LCM) of two numbers. Students examine the rows and columns of an area model to identify multiples and the LCM. They describe the relationship between the product, GCF, and LCM.

• Prime factorization is a method to determine common factors and common multiples of two numbers.

• The greatest common factor (GCF) of two numbers is the largest factor shared by the two numbers.

• The least common multiple (LCM) of two numbers is the smallest non-zero multiple shared by the two numbers

• The Commutative and Distributive Properties are properties used to generate equivalent expressions.

• If two numbers a and b are relatively prime, then the GCF (a, b) = 1 and the LCM (a, b) = ab.

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3

Composing and Decomposing NumbersLeast Common Multiple and Greatest Common Factor

Students continue to expand their understanding of factors, multiples, common factors, and common multiples as introduced in previous lessons. They use greatest common factor (GCF) and least common multipe (LCM) to solve problems.

• Number relationships are useful in solving problems in context.

• Common factors help determine how to divide or share things equally.

• Common multiples help determine how things with different cycles can occur at the same time.

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Topic 2: Positive Rational NumbersELPS: 1.A, 1.C, 1.E, 1.F, 1.H, 2.C, 2.D, 2.E, 2.I, 3.D, 3.E, 3.F, 3.G, 3.H, 4.F, 4.G, 5.F


1Rocket StripsDividing a Whole into Fractional Parts

Students create fraction strips for unit fractions 1 __ 2 , 1 __

3 , 1 __

4 , 1 __

6 , 1 __

8 , 1 ___

12 ,

and 1 ___ 16

. They identify equivalent fractions by aligning the fraction strips on the fold lines, and then complete a graphic organizer to represent all the equivalent fractions represented by the fraction strips. Students conclude that the numerator and denominator of equivalent fractions are multiples of the original unit fractions.

• Fraction strips are used to compare fractions with different denominators.

• A unit fraction is a fraction that has a numerator of 1 and a denominator that is a positive integer.

• Equivalent fractions are fractions that represent the same part-to-whole relationship.

• Equivalent fractions are fractions generated by multiplying both the numerator and denominator by the same factor.

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2Getting CloserBenchmark Fractions

Students translate their understanding of fraction strips to number lines. They use the benchmark fractions 0, 1 __

2 , and 1 to

estimate the value of fractions, write fractions that are close to these benchmarks, and estimate sums. Students solve a problem which involves comparing fractions that represent shaded parts of figures.

• Benchmark fractions are common fractions used to estimate the value of fractions such as 0, 1 __

2 , and 1.

• A fraction is close to 0 when the numerator is very small compared to the denominator.

• A fraction is close to 1 __ 2 , when the numerator is

about half the size of the denominator.• A fraction is close to 1 when the numerator is very

close in size to the denominator.

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3Did You Get the Part?Multiplying Fractions

Students review the area model for multiplication and apply it to multiplying mixed numbers. They analyze two methods for multiplying mixed numbers and then use these methods to answer questions in the context of a real-world scenario.

• Area models can be used to illustrate the multiplication of two fractions, which is essentially the same as taking a part of a part.

• An area model representing the multiplication of two mixed numbers can be tiled with fractional unit squares to express the product as an improper fraction.

• The product of two fractions represented by an area model is the same as the product of the fractions calculated using the standard algorithm.

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4

Yours IS to Reason Why!Fraction by Fraction Division

Students connect multiplication to division by writing fraction fact families for area models. They then use fraction strip and number line models to investigate the division of fractions by fractions. Students use these models to develop an algorithm for rewriting division sentences as multiplication sentences. They apply the procedure to solve problems involving fractions and mixed numbers.

• Area models and fact families can be used to illustrate the quotients of fractions.

• The reciprocal or multiplicative inverse of a number a—b is the number b—a where a and b are nonzero numbers.

• To calculate the quotient of two fractions, multiply the dividend by the reciprocal of the divisor.

• There are other algorithms to divide fractions, such as dividing across in special cases and using complex fractions as a form of 1.

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Topic 3: Shapes and SolidsELPS: 1.A, 1.C, 1.E, 1.F, 1.G, 2.C, 2.E, 2.I, 3.D, 3.E, 4.B, 4.C, 5.B, 5.F, 5.G


1Consider Every SideConstructing Triangles Given Sides

Students use patty paper, pasta, and construction tools to explore the information required to create no triangles, unique triangles, or multiple triangles when given two or three possible side lengths. They learn that an infinite number of triangles can be made from only two side lengths. They also learn that unique triangles are formed when provided with three segments that are sufficiently long in relation to each other. Students should note that if all the measures of a triangle are the same as another triangle, even though they are in different orientations, the provided information creates a unique triangle. Students then summarize their knowledge of the conditions that form 0, 1, or multiple triangles.

• Constructing a triangle given the length of two sides does not result in the construction of a unique triangle.

• Constructing a triangle given the length of three segments, such that the sum of two segment lengths is greater than the third length, results in the construction of a unique triangle.

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2Turning a One-Eighty!Triangle Sum Theorem

Students explore and justify the relationships between angles and sides in a triangle. They establish the Triangle Sum Theorem and use the theorem as they explore the relationship between interior angle measures and the side lengths of triangles. They then practice applying the theorem.

• The Triangle Sum Theorem states that the sum of the measures of the interior angles of a triangle is 180°.

• The longest side of a triangle lies opposite the largest interior angle.

• The shortest side of a triangle lies opposite the smallest interior angle.

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3

All About That Base… and HeightArea of Triangles and Quadrilaterals

Students use previously known area formuals and the principle of area conservation to investigate the areas of parallelograms, triangles, and trapezoids. They use this knowledge to develop formulas for the areas of these shapes, practice calculating areas, and solving area-related problems. Students learn that the choice of base or height does not affect the area of the shape.

• The formula for the area of a rectangle is A = lw, where A is the area of the rectangle, l is the length of the rectangle, and w is the width of the rectangle.

• The formula for the area of a parallelogram is A = bh, where A is the area of the parallelogram, b is the length of the base of the parallelogram, and h is the height of the parallelogram.

• The formula for the area of a triangle is A = 1—2 bh, where A is the area of the triangle, b is the length of the base of the triangle, and h is the height of the triangle.

• The formula for the area of a trapezoid is A = 1—2 h (b1 + b2 ), where A is the area of the trapezoid, h is the height of the trapezoid, and b1 and b2 are bases of the trapezoid.

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4

Length, Width, and DepthDeepening Understanding of Volume

In this lesson, students recall that they can calculate the volume of rectangular prisms using V=lwh and V=Bh. They pack prisms with fractional side lengths using fractional unit cubes and then multiply the number of cubes by the volume of each unit cube. Students then practice solving real-world problems, including packing problems.

• A unit cube is a cube that is one unit in length, one unit in width, and one unit in height.

• Volume is the amount of space occupied by an object.

• The formula for the volume of a cube is V = l • w • h, where l is the length, w is the width and h is the height, or V = b • h, where B is the area of the base and h is the height.

• Another way to calculate volume is to pack prisms using fractional unit cubes and then multiply the number of cubes by the volume of each unit cube.

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Topic 4: DecimalsELPS: 1.A, 1.B,1.C, 1.D, 1.E, 1.F, 1.G, 1.H, 2.C, 2.D, 2.E, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.E, 3.F, 3.G, 3.J, 4.A, 4.B, 4.C, 4.D, 4.F, 4.G, 4.I, 4.K, 5.A, 5.B, 5.C, 5.D, 5.E, 5.F, 5.G


1You Have a PointPlotting, Comparing, and Ordering Rational Numbers

In this lesson, students investigate place value by using a human number line to plot decimal values. They plot given decimals on a number line and identify other decimals that lie between them. Students create a rule to compare decimals and apply their rule in context. They use a number line to compare decimals and fractions.

• A decimal is a number written in a system based on multiples of 10 and is another way to represent parts of a whole.

• You can plot any decimal value on a number line by determining between which two known values it lies.

• There is always a value between any two points on a number line.

• When comparing two decimal values, rewrite them so that they have the same number of decimal places.

• When comparing a fraction and a decimal, consider their placements on a number line.

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2Get in LineAdding and Subtracting Decimals

In this lesson, students use place value to estimate sums and differences of decimals and then develop standard algorithms. They solve real-world problems by first determining whether they need to add or subtract, using estimation to predict the magnitude of the answer, and then applying the standard algorithm. Students also play a calculator game to target place value in subtraction.

• Estimating the decimal sum or difference before completing a calculation is a useful strategy to check if the actual answer is reasonable.

• To add or subtract decimals, align the digits with like place values by first lining up the decimal points.

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3Product PlacementMultiplying Decimals

In this lesson, students use an area model on a hundredths grid to represent the multiplication of two decimals less than one. They use estimation to reason about the placement of the decimal point in multiplication problems and then analyze patterns to develop the algorithm for multiplying decimals. Students solve area and volume problems that require multiplying, adding, and subtracting decimals.

• An area model using a hundredths grid can represent the product of two decimals less than one.

• Use estimation to determine if the product of two decimal factors is reasonable.

• When multiplying decimals, the number of decimal places in the product is equal to the sum of the decimal places in the factors.

• You can use the standard algorithms for decimal addition, subtraction, and multiplication to solve real-world problems.

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4Dividend in the HouseDividing Whole Numbers and Decimals

In this lesson, students use the standard algorithm for long division with whole numbers. They demonstrate how the algorithm works for decimal dividends by relating it to a model and make sense of why the algorithm is modified to accommodate decimal divisors. Students solve area and volume problems requiring decimal division.

• The long division algorithm is based on an organized estimation process to determine the quotient.

• When a quotient has a remainder, the situation informs how to interpret the remainder.

• When you have a decimal divisor, multiply it by a power of ten to convert it to a whole number. Then, multiply the dividend by the same power of ten. Because you multiplied both the dividend and divisor by the same power of ten, the quotient will be the same as the quotient of the original problem.

• You can use the standard algorithms for whole number and decimal division to solve real-world problems.

• Use estimation to determine if the quotient of a division problem is reasonable.

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2 Relating QuantitiesTopic 1: RatiosELPS: 1.A, 1.C, 1.D, 1.E, 2.C, 2.D, 2.G, 2.H, 3.A, 3.B, 3.C, 3.D, 3.E, 3.F, 4.A, 4.B, 4.C, 4.F, 4.K, 5.E, 5.F


1It's All RelativeIntroduction to Ratio and Ratio Reasoning

Students differentiate between additive and multiplicative reasoning in preparation for the study of ratios. The term ratio is defined as a comparison between two quantities; ratios employ multiplicative reasoning. Students compare quantities using part-to-part and part-to-whole ratios. They write ratios in words, in colon notation, and in fractional form. They identify fractions and percents as special types of part-to-whole ratios.

• A ratio is a comparison of two quantities.• Ratios can be expressed using words, with a colon,

or in fractional form.• A ratio can represent part-to-whole or part-to-part

relationships.• Fractions and percents are special types of part-to-

whole ratios.

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2Going StrongComparing Ratios to Solve Problems

Students explore ratios in different real-world situations. They decide which of two or more ratios in each situation is greater using qualitative and quantitative reasoning. Students compare part-to-part and part-to-whole ratios represented pictorially, verbally, and numerically. The focus in this lesson is on reasoning rather than on computation.

• A ratio is a comparison of two quantities.• Qualitative comparisons are made in the absence

of numeric values.

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3

Oh, Yes, I Am the Muffin ManDetermining Equivalent Ratios

Students are introduced to formal strategies to determine equivalent ratios, including pictures, tape diagrams, scaling up/down, and double number lines. They solve a variety of real-world problems using these strategies to create equivalent equations. An example of scaling up ratios is provided, and students use the example to answers questions in various contexts. The definitions of scaling up, scaling down, and scale factor are provided. Students then determine equivalent ratios by either scaling up or scaling down both parts of the ratio by a scale factor. A double number line is introduced. They use double number lines to represent the proportional relationship between two quantities and solve for unknown quantities.

• Models are used to represent ratio relationships and to solve real-world problems.

• A ratio is a comparison of two quantities.• A rate is a ratio that compares two quantities that

are measured in different units.• When two rates or ratios are equal to each other,

they can be written as a proportion.• A proportion is an equation that states two ratios

are equal.• When writing a proportion, the numbers

representing the same quantity must be placed in both numerators or in both denominators. The unit of measurement must be consistent among the ratios.

• Scaling up means to multiply both part of a ratio by the same scale factor greater than 1, or divide both parts of a ratio by the same scale factor less than 1.

• Scaling down means to divide both parts of a ratio by the same factor greater than 1, or multiply both parts of a ratio by the same scale factor less than 1.

• A double number line is a model that is made up of two number lines used to represent the equivalence of two related numbers. The intervals on each number line maintain the same ratio.

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4A Trip to the MoonUsing Tables to Represent Equivalent Ratios

Students use tables in different ways to determine equivalent ratios. They multiply or divide existing ratios by a common factor to determine equivalent ratios in a table, just as they did in scaling. Students learn that existing ratios in a ratio table can be added to form new equivalent ratios. They then complete equivalent ratio tables for different proportional situations.

• Ratios are used to represent proportional relationships in the real world.

• Equivalent ratios are generated within the context of a situation using addition, subtraction, multiplication, and division.

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5They're Growing!Graphs of Ratios

Students investigate rectangles with a common ratio of side lengths and those with a constant difference in side lengths. They graph the dimensions of the rectangles on a coordinate plane and conclude that equivalent ratios represented on the coordinate plane form a straight line that passes through the origin. Students analyze a ratio that is represented using a table, double number line, and coordinate plane. The models are connected and used to solve real-world problems.

• Equivalent ratios can be represented by tables, double number lines, and on coordinate planes.

• A ratio y—x is plotted as the ordered pair (x, y).• Equivalent ratios represented on the coordinate

plane form a straight line that passes through the origin.

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6One is Not EnoughUsing and Comparing Ratio Representations

Graphs and double number lines of real-life situations are given. Students interpret the points on the graphs in terms of the problem situation. They determine unknown ratios using either a specified strategy or the strategy of their choice. Students also contrast representations of additive and multiplicative relationships. Students then create a graphic organizer to show how equivalent ratios can be modeled through four representations: scale up/scale down, tables, double number lines, and graphs.

• Equivalent ratios represented by tables, double number lines, and on coordinate planes can be used to solve real-world and mathematical problems.

• Equivalent ratios represented on the coordinate plane form a straight line that passes through the origin.

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Topic 2: PercentsELPS: 1.A, 1.C, 1.D, 1.E, 2.C, 2.D, 2.G, 2.H, 3.A, 3.B, 3.C, 3.D, 3.E, 3.F, 4.A, 4.B, 4.C, 4.F, 4.K, 5.E, 5.F


1We Are Family!Percent, Fraction, and Decimal Equivalence

Students learn about the relationships between percents, fractions, and decimals. In the first activity, students analyze the results of a survey of one hundred students. They complete a table by writing the ratio, fraction, and decimal equivalences for each result. Students use hundredths grids to model the result, and they then write the percent equivalence. They are reminded that percents are special types of part-to-whole ratios. A percent is described as a fraction in which the denominator is 100 and the % symbol represents the phrase “out of 100.” Students write numbers in equivalent forms and use number lines to indicate the equivalent fraction, decimal, and percent represented by the markers on the number line. They analyze reasoning about combining ratios into an overall percent. They then play a percentage match game to identify equivalent representations. A chart is provided in the summary to highlight common fraction, decimal, and percent equivalents.

• Percent is a part-to-whole ratio with a whole of 100. The symbol "%" means “out of 100.”

• The hundredths grid can be used to represent a fraction, decimal, or percent.

• To write a fraction as a percent, scale up or down to an equivalent fraction with a denominator of 100, if possible.

• To write a fraction as a percent, divide the numerator by the denominator and move the decimal in the quotient two places to the right.

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2Warming the BenchUsing Estimation and Benchmark Percents

Students begin the lesson building fluency with ordering fractions, decimals, and percents. They then estimate the percent of cylinders, circles, and squares that are partially shaded. Students write estimates as fractions, decimals, and percents. Benchmark percents are introduced to help students mentally estimate the value of a percent. They then use calculators to investigate the values of 1% and 10% of several numbers. Students write rules about moving the decimal two places to the left to determine 1% of any number and moving the decimal one place to the left to determine 10%. Various scenarios are presented in which students are asked to estimate and calculate percents.

• Percent is a fraction in which the denominator is 100. The symbol "%" means “out of 100.”

• A benchmark percent is a percent that is commonly used, such as 1%, 5%, 10%, 25%, 50%, and 100%.

• Calculating 1% of any number is the same as moving the decimal point two places to the left.

• Calculating 10% of any number is the same as moving the decimal point one place to the left.

• Benchmark percents can be used to perform mental estimation and calculation of percents.

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3

The Forest for the TreesDetermining the Part and the Whole in Percent Problems

Percent problems involve three quantities: the part, the whole, and the percent. In this lesson, students solve for the percent, given the part and the whole, and solve for the whole, given the percent and the part. They set up a proportion where the percent, if known, is written as a fraction with a denominator of 100. They then determine the unknown, using multiplication.

• Percent problems involve three quantities: the part, the whole, and the percent.

• When calculating the whole, given the percent and the part, write the percent as a fraction with a denominator of 100 and set it equal to the part over x. Then solve for x.

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Topic 3: Unit Rates and ConversionsELPS: 1.A, 1.C, 1.D, 1.E, 2.C, 2.D, 2.G, 2.H, 3.A, 3.B, 3.C, 3.D, 3.E, 3.F, 4.A, 4.B, 4.C, 4.F, 4.K, 5.E, 5.F

1

Several Ways to MeasureUsing Ratio Reasoning to Convert Units

Students deepen their understanding of converting units of measurement through the use of ratio reasoning and strategies for determining equivalent ratios. The term convert is defined, and students use approximate conversion rates to estimate measurement conversions before engaging with formal methods of converting. Converting among units of measurement in the same system is recast in terms of conversion ratios, which can also be called conversion rates.

Students use ratio reasoning and strategies to convert within the U.S. customary system and the metric system. Students use double number lines, ratio tables, and scaling up and down to convert units of measurement. They analyze Worked Examples of the different strategies. For scaling up and down, students explain why one conversion ratio is more appropriate than the other. Finally, students are introduced to unit or dimensional analysis as a strategy for converting between units of measurement. They practice using unit analysis in problems about distance, money, and area. Students make choices about which strategy to use when converting between units of measurement.

• When a smaller unit of measure is converted to a larger unit of measure, the larger unit of measure has fewer units.

• When a larger unit of measure is converted to a smaller unit of measure, the smaller unit of measure has more units.

• All of the strategies used to determine equivalent ratios (double number lines, ratio tables, scaling up and down) can be used to convert between units.

• Unit analysis is a strategy for converting units that ensures the correct calculations and units in the final result.

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2What Is the Best Buy?Introduction to Unit Rates

Unit rates are introduced. Students utilize models to estimate unit rates two different ways. They compare the different methods and conclude that both methods lead to correct solutions. Students write unit rates that compare the same quantities in two different ways. They then use unit rates to determine the best buy. Students compute unit rates to make comparisons about gas mileage, loaves of bread per person at a dinner, speed of runners, baking times, cafeteria milk sales, and the speed of buses. Finally, they complete problems about constant speeds and determining multiple numbers of various items.

• A rate is a ratio in which the two quantities being compared are measured in different units.

• A unit rate is a comparison of two measurements in which the denominator has a value of one unit.

• Unit rates are used to calculate best buys.• Unit rates are used to make comparisons involving

rates.

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3

Seeing Things DifferentlyMultiple Representations of Unit Rates

Students use what they know about unit rates to further develop flexible thinking and problem solving with unit rates in different situations using a variety of representations, including tables and graphs. The lesson begins with students investigating a speedometer as a double number line. Students then reason with unit rates in various mathematical and real-world situations, including measuring the diagonals of a Golden Rhombus and investigating the speed of the Duquesne Incline. Finally, students demonstrate their learning by creating a situation of their own to represent the graph of equivalent rates.

• Equivalent rates can be represented through tables, double number lines, and on coordinate planes.

• Points on a straight line that passes through the origin describe equivalent rates.

• Unit rates are helpful when making comparisons.

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3 Moving Beyond Positive QuantitiesTopic 1: Signed Numbers and the Four QuadrantsELPS: 1.A, 1.C, 1.D, 1.E, 1.G, 2.C, 2.D, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 4.A, 4.B, 4.C, 4.D, 4.G, 4.K, 5.C, 5.D, 5.E


1Human Number LineIntroduction to Negative Numbers

Students extend their knowledge of number to the negatives by building on prior knowledge of ordering positive rational numbers and plotting them on a number line. Students learn that opposite on a number line means to reflect over the origin. They also learn that the negative sign is used as notation for opposites. Students explain the meaning of 0, positive numbers, and negative numbers in a variety of contexts.

• Positive and negative numbers describe quantities having opposite directions or values.

• Positive and negative numbers are used in real-world situations.

• Zero has different meanings in different real-world situations.

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2Magnificent MagnitudeAbsolute Value

Students formalize the idea that opposites are the same distance from zero and call this distance the absolute value of a number. Students continually revisit the meaning of absolute value, focusing on distance from 0. Students evaluate absolute value statements and compare numbers using absolute values. Students solve problems using absolute value statements.

• The distance from zero is the absolute value, or magnitude, of a rational number.

• Absolute values are used to describe real-world situations.

• Absolute value equations are used to compute distance on a number line.

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3What's in a Name?Rational Number System

Students formally classify numbers as rational numbers and understand that all numbers they have studied so far are subsets of the rational numbers. Students sort and classify numbers. They investigate the density of rational numbers by locating rational numbers between other rational numbers.

• Rational numbers are the set of numbers that can written as a—b where a and b are integers and b does not equal 0.

• The set of rational numbers includes the sets of integers, whole numbers, and natural numbers.

• Given two rational numbers, there exists an infinite number of rational numbers between those numbers.

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4

Four Is Better Than OneExtending the Coordinate Plane

Students build from working with rational numbers on a number line to rational numbers on a coordinate plane. They identify the four quadrants, identify points, and make generalizations about points located in given quadrants. Students determine distances between two points that have a common coordinate.

• The coordinate plane is used to plot ordered pairs of rational numbers.

• The coordinate plane has 4 quadrants that are named with Roman numerals.

• The relationship between two ordered pairs differing only by signs is a reflection across one or both axes.

• Absolute value equations are used to determine the distance between two points that share an x-coordinate or a y-coordinate.

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Topic 2: Operating with IntegersELPS: 1.A, 1.C, 1.D, 1.E, 1.G, 2.C, 2.D, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 4.A, 4.B, 4.C, 4.D, 4.G, 4.K, 5.E


1

Math FootballUsing Models to Understand Integer Addition

A math football game is used to model the sum of positive and negative integers. Rules for the game and a game board are provided. Students use number cubes to generate the integers. They then take that same information and write integer number sentences.

• A model can be used to represent the sum of a positive and negative integer, two negative integers, or two positive integers.

• Information from a model can be rewritten as an equation.

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2Walk the LineAdding Integers, Part I

A number line is used to model the sum of two integers. Students begin the lesson by walking a number line on the floor of the classroom. Through a series of activities, students will notice patterns for adding integers. After the kinesthetic activity, students examine a Worked Example and then practice calculating sums of positive and negative numbers using a number line model. Questions focus students on the distance an integer is from 0 on the number line, or the absolute value of the integer, to anticipate writing a rule for the sum of two integers having different signs. Students demonstrate their understanding of the patterns by writing informal rules for adding integers. Finally, they use a number line model to determine unknown values in equations.

• On a number line, when adding a positive integer, move to the right.

• On a number line, when adding a negative integer, move to the left.

• When adding two positive integers, the sign of the sum is always positive.

• When adding two negative integers, the sign of the sum is always negative.

• When adding a positive and a negative integer, the sign of the sum is the sign of the number that is the greatest distance from zero on the number line.

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3Two-Color CountersAdding Integers, Part II

Through a series of activities with two-color counters, students will develop rules for adding integers. Students determine that to have a sum of zero, two integers must have opposite signs but the same absolute value. Examples of modeling the sum of two integers with opposite signs using two-color counters are provided. The counters are paired together, one positive counter with one negative counter, until no possible pairs remain. The resulting counters determine the sum of the integers. Several models are given and students write a number sentence to represent each model. Students critique reasoning about using the two-color counters to model adding integers. They draw models for given number sentences and create number sentences for given models. They create a graphic organizer to represent the sum of additive inverses using a variety of representations.

• Opposite quantities in real-life situations combine to make 0.

• Two numbers with the sum of zero are called additive inverses.

• When two integers have the same sign and are added together, the sign of the sum is the sign of both integers.

• When two integers have opposite signs and are added together, the integers are subtracted and the sign of the sum has the sign of the integer with the greater absolute value.

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4What's the Difference?Subtracting Integers

Number lines and two-color counters are used to model subtraction of signed numbers. Through a series of activities, students will develop rules for subtracting integers. As in the lesson on adding signed numbers, the number line method is used to model the difference between two integers. Students then learn how to use zero pairs when performing subtraction using the two-color counter method. Students analyze real-world situations that require calculating the distance between two signed numbers. They build on what they already know about absolute value to determine the distance.

• Subtraction of integers can be modeled using a number line and two-color counters.

• Subtracting two negative integers is similar to adding two integers with opposite signs.

• Subtracting a positive integer from a positive integer is similar to adding two integers with opposite signs.

• Subtracting a positive integer from a negative integer is similar to adding two negative integers.

• Subtracting two integers is the same as adding the opposite of the subtrahend, the number you are subtracting.

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5Equal GroupsMultiplying and Dividing Integers

Two-color counters and number lines are used to model the product of two integers. Through a series of activities, students develop rules to determine the sign of a product or quotient of two integers. They conclude that multiplying or dividing two positive integers or two negative integers always results in a positive product or quotient, and that multiplying or dividing a positive integer by a negative integer always results in a negative product or quotient. Questions focus students on the sign of a product resulting from the multiplication of two positive integers, two negative integers, and one positive and one negative integer. Students apply this knowledge to determine the sign of the product that results from multiplying three or more integers.

• Multiplication of integers can be modeled using a number line or two-color counters.

• The product that results from multiplying two positive integers is always positive.

• The product that results from multiplying two negative integers is always positive.

• The product that results from multiplying a negative integer and a positive integer is always negative.

• The product that results from multiplying an odd number of negative integers is always negative.

• The product that results from multiplying an even number of negative integers is always positive.

• Division and multiplication are inverse operations.

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4 Determining Unknown QuantitiesTopic 1: ExpressionsELPS: 1.A, 1.C, 1.D, 1.E, 1.G, 2.C, 2.D, 2.G, 2.H, 2I, 3.A, 3.B, 3.C, 3.D, 3.F, 4.A, 4.B, 4.C, 4.D, 4.G, 4.K, 5.B, 5.C, 5.E


1Relationships MatterEvaluating Numeric Expressions

Students write and simplify numeric expressions. The terms power, base, and exponent are revisited, and the terms perfect square, perfect cube, and Order of Operations are introduced and defined. Students create numeric expressions to represent geometric models and draw geometric models to represent numeric expressions. Students learn that an expression represents a relationship between quantities, rather than a recipe to perform operations on values. Students conclude the lesson by applying the Order of Operations to simplify numeric expressions.

• A numeric expression is a mathematical phrase containing numbers.

• To simplify a numeric expression means to calculate an expression to get a single value.

• Parentheses are symbols used to group numbers and operations, and they are used to change the normal order in which operations are performed.

• The Order of Operations is a set of rules that ensures the same result every time an expression is simplified.

1. Simplify expressions inside parentheses orgrouping symbols such as ( ) or [ ].

2. Simplify terms with exponents.3. Multiply and divide from left to right.4. Add and subtract from left to right.

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2Into the UnknownIntroduction to Algebraic Expressions

Students write algebraic expressions and evaluate numeric expressions. They practice writing algebraic expressions for mathematical word sentences, and then reverse the process. Students decompose given algebraic expressions by stating the number of terms in each algebraic expression and listing the terms. Students conclude the lesson by evaluating algebraic expressions individually and in table form. Finally, they practice composing algebraic expressions from verbal phrases written with mathematical terminology.

• A variable is a letter or symbol used to represent quantities.

• An algebraic expression is a mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

• Situations can be expressed using algebraic expressions.

• A numerical coefficient is a number, or quantity, that is multiplied by a variable in an algebraic expression.

• If a variable does not have a coefficient, then it is understood to be 1.

• A constant is a number, or quantity, that does not change its value.

• To evaluate an algebraic expression, substitute the given values for the variables and then apply the Order of Operations rules to the numerical expression.

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3Second Verse, Same as the FirstEquivalent Expressions

Students consider a situation about packing two suitcases for a camping trip and then combining the contents of the suitcases to motivate the need to combine like terms in algebraic expressions. Students model and simplify algebraic expressions first by using algebra tiles to make sense of combining like terms and then by using the rules and properties. Algebra tiles are then used as a method to make sense of the Distributive Property. Students rewrite expressions using the Distributive Property, the Order of Operation Rules, and combining like terms. Then students use algebra tiles to apply the distributive property to division problems. Finally, students rewrite expressions as a product of two factors.

• Algebra tiles are a helpful tool to make sense of rewriting algebraic expressions.

• Like terms are two or more terms that have the same variable raised to the same power.

• The Distributive Property states that if a, b, and c are any real numbers, then a(b + c) = ab + ac. Because subtraction is a special form of addition and division is a special form of multiplication, the Distributive Property can also be expressed as a(b 2 c) 5 ab 2 ac,

a 1 b

_____ c 5

a

__ c 1

b

__ c , and

a 2 b _____ c 5

a

__ c 2

b

__ c .

• An algebraic expression can be written as the product of two factors by applying the Distributive Property.

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4

Are They Saying the Same Thing?Verifying Equivalent Expressions

Students begin by reviewing the properties of numbers and operations that they have formally or informally studied in the past. This allows students to use properties as they rewrite algebraic expressions in equivalent forms. Students analyze pairs of expressions. They use properties, tables, and graphs to show that the expressions are or are not equivalent. Students compare the algebraic expressions and are asked to use tables and graphs to determine if they equivalent. This opens the discussion that one non-example is necessary to disprove a claim, while an infinite number of examples are necessary to prove a claim.

• The Commutative Properties of Addition and Multiplication state that the order in which you add or multiply two or more numbers does not affect the sum or use the product.

• The Associative Properties of Addition and Multiplication state that changing the grouping of the terms in an addition or multiplication problem does not change the sum or product.

• The Distributive Property states that if a, b, and c are any real numbers, then a(b + c) = ab + ac. Because subtraction is a special form of addition and division is a special form of multiplication, the Distributive Property can also be expressed as a(b 2 c) 5 ab 2 ac,

a 1 b

_____ c 5

a

__ c 1

b

__ c , and

a 2 b

_____ c 5

a

__ c 2

b

__ c .

• Two algebraic expressions are equivalent expressions if, when the same value is substituted for the variable into each expression, the results are equal.

• Two algebraic expressions can be proven to be equivalent by: (1) using the properties of numbers and operations to simplify them until they are written the exact same way; and (2) graphing each expression on the same graph to determine if their graphs are the same.

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5

DVDs and SongsUsing Algebraic Expressions to Analyze and Solve Problems

Students practice writing algebraic expressions and using those expressions to solve problems. In the first problem, they write four different sets of algebraic expressions to represent the same situation, each time basing their expressions upon a different initial variable representing a different varying quantity in the problem. Students use the algebraic expressions to answer other questions about the same situation. The second problem is set up the same way as the first problem, the only differences being the context, the number of relationships described in the problem, a minor change in the way the relationships are described, and the number of algebraic expressions students are asked to write. Students then use equivalent expressions to explain why a number trick works and then they create their own number trick.

• Many real-life situations can be represented using algebraic expressions. The algebraic expressions can then be used to answer questions about the situation.

• Different algebraic expressions may represent the same real-life situation depending upon what the initial variable represents.

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Topic 2: Equations and InequalitiesELPS: 1.A 1.D, 1.E, 1.G, 2.C, 2.D, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 3.H, 3.I, 4.A, 4.B, 4.C, 4.G, 4.K, 5.E


1First Among EqualsReasoning with Equal Expressions

Students learn that an equation is a mathematical sentence created by equating two expressions. They create equations from a list of expressions and determine the solutions to their equations using substitution. Students learn that equations may have one solution, no solution, or infinite solutions. Students use Properties of Equality to write equations that have the same solution as a given equation. They identify the Zero Property of Multiplication, the Identity Property of Multiplication, and the Identity Property of Addition. Students are introduced to algebraic inequalities by analyzing their graphs and solution sets, including inequalities of the form x > c and x < c. Students write inequalities represented on a number line and graph the solution sets of other algebraic inequalities. Then, they consider how many solutions an inequality may have.

• A solution to an equation is any value for a variable that makes the equation true.

• The Properties of Equality state that if the same operation is performed on both sides of the equation, then equality is maintained.

• The graph of an inequality in one variable is the set of all points on a number line that make the inequality true.

• The solution set of an inequality is the set of all points that make the inequality true.

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2Bar NoneSolving One-Step Addition Equations

Students use bar models to solve a variety of one-step addition equations. They analyze Worked Examples and analyze solution strategies to develop an understanding of using bar models to solve addition equations. The term inverse operations is defined. Students use properties of arithmetic and algebra to solve addition equations without using models. They eventually use the Subtraction Property of Equality to solve a variety of addition equations where the solutions also include integers. Finally, students summarize how to solve and check one-step addition equations.

• A one-step equation is an equation that can be solved using only one operation.


• To solve an equation, you must isolate the variable by performing inverse operations.

• The Properties of Equality state that if you perform the same operation on both sides of an equation, then equality is maintained.

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3Play It In ReverseSolving One-Step Multiplication Equations

Students reason about and solve a variety of one-step multiplication equations of the form px = q, where p, x, and q are non-negative rational numbers. They first analyze Worked Examples and create bar models to understand the structure of equations in this form and reason about their solutions. Through composition and decomposition to isolate the variable using bar models, students are primed to formalize their strategies using inverse operations and Properties of Equality. Students then solve multiplication equations without using models and provide justification for their solution strategies. Students then solve multiplication equations without using models and provide justification for their solution strategies. Finally, students analyze a set of equations and determine the most efficient solution strategy based on the form of the multiplication equation.



• To solve an equation, you must isolate the variable using Properties of Equality.


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4The Real DealSolving Equations to Solve Problems

Students solve a variety of real-world and mathematical problems that can be modeled by one-step equations. They are introduced to literal equations and use the skills learned in the previous lesson to solve them. Students are then presented with a set of direct statement problems as a way to introduce a mathematical structure (defining variables, writing an equation, solving the equation, and interpreting the solution) to solve real-world problems. This activity is followed by a set of problems that are not as straightforward in nature, which require the use of area formulas.

• A mathematical framework can be used to solve real-world problems.

• Variables can be used to represent quantities in expressions describing real-world values.

• Equations can be used to model relationships between variables.


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5Greater Than MostSolving Inequalities with Inverse Operations

Students solve inequalities and graph the solutions on number lines. They use empirical examples to informally state the Properties of Inequalities. Students solve a variety of one-step inequalities. Attention is also given to verifying the solution to an inequality. Finally, students write their own real-world scenarios given three inequality statements.

• An inequality is any mathematical sentence that has an inequality symbol such as >, <, $, or #.

• The graph of an inequality with one variable is the set of all points on a number line that make the inequality true.

• The solution set of an inequality is the set of all points that make the inequality true.

• The inequality symbol remains the same when adding, subtracting, multiplying, or dividing an inequality by a positive number.

• The inequality symbol reverses when multiplying or dividing an inequality by a negative number.

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Topic 3: Graphing Quantitative RelationshipsELPS: 1.A, 1.C, 1.D, 1.E, 1.G, 2.C, 2.D, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 4.A, 4.B, 4.C, 4.D, 4.G, 4.K, 5.E


1

Every Graph Tells a StoryIndependent and Dependent Variables

Students create scenarios to match numberless graphs in which the axes are labeled.They then cut out graphs and match them with the appropriate scenario. Students will then label the axes and analyze the graphs based on their prior knowledge, including ratio relationships and using inequality statements to represent constraints in problem situations. They determine how one quantity depends on another using scenarios, equations, and graphs. Students then identify independent and dependent quantities and represent those quantities using variables. They write an equation, complete a table of values, and create a graph to model the situation. Finally, students analyze two situations in which the independent and dependent quantities are reversed to understand that the question being asked often determines which is the independent quantity and which is the dependent quantity.

• Graphical representations are used to solve problems.

• Graphs represent the relationships between independent and dependent quantities.

• •When one quantity is determined by another in the problem situation, it is said to be the dependent quantity. The varaible representing the dependent quantity is the dependent variable.

• When one quantity is not determined by another in the problem situation, it is said to be the independent quantity. The variable representing the independent quantity is the independent variable.

• The independent variable is located on the x-axis and the dependent variable is located on the y-axis.

• When writing an equation, it can be helpful to isolate the dependent variable to more clearly see the relationship between quantities.

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2

The Power of the Horizontal LineUsing Graphs to Solve Problems

Students determine unknown values for a scenario and use the values to write a one-step multiplication equation to represent the scenario. They then analyze the graph of the equation, interpreting ordered pairs on the graph and determining the unit rate. Students use the graph to determine the value of an independent quantity using a horizontal line graphed at the value of the dependent variable. They then answer a variety of questions about the scenario; some answers are single solutions and some include a range of values (inequality statements).

Given a scenario, students analyze the graph of the scenario and write an equation to represent the scenario. Students then must decide when to use the graph and when to use the equation to answer a variety of questions about the scenario. Finally, students compare equations and graphs that represent additive and multiplicative relationships. Students write equations for the graphs and explain how to use a graph to solve one-step equations.

• Multiple representations such as words, tables, equations, and graphs are used to solve problems of the form y = x + b and y = kx for cases in which y, x, and b are all non-negative rational numbers.


• A solution to an equation represented on a graph is any point on the line.

• An inequality of the form x > c or x < c can be used to represent constraints when solving a real-world problem.

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3

Planes, Trains, and PaychecksMultiple Representations of Equations

In this lesson, students analyze equations in a variety of different forms—represented in tables, graphs, in word problems, and as algebraic equations. They solve problems using these multiple representations of equations. Students continue to explore discrete and continuous quantities.

• Multiple representations, such as words, tables, equations, and graphs are used to solve problems.

• Graphs can be characterized as being continuous or discrete based upon the scenario they model and the units of the independent and dependent variables.

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4

Time for Triathlon TrainingRelating Distance, Rate, and Time

Students analyze and solve problems about competing in triathlons to investigate the relationship between distance, rate, and time. In each activity, students analyze the rate for a specific segment of the triathlon. Each activity begins with either a graph, table, or given rate. Students recognize that the equation d = rt, where d represents the distance traveled, r represents the rate of the distance traveled to the time, and t represents the time, was used in each activity. They summarize the relationship between distance, rate, and time, and they use the equation to solve a variety of real-world problems.

• Graphical representations are used to solve problems.

• Multiple representations such as words, tables, equations, and graphs are used to solve problems of the form y = kx for cases in which y, k, and x are all non-negative rational numbers.

• The equation d = rt, where d represents the distance traveled, r represents the rate of the distanced traveled to the time, and t represents the time, can be used to solve a variety of real-world problems.

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5

There Are Many Paths...Problem Solving on the Coordinate Plane

Students apply their knowledge of plotting and interpreting rational numbers on the coordinate plane, creating tables of values, and writing and solving equations to solve a variety of problems. They model real-life situations, analyze data, and select which representation to use for specific problems.

• Multiple representations such as situations written in words, equations, tables, and graphs can be used to solve problems.

• Graphs other than lines can be used to model real-life situations.

• Graphs can be used to interpret data and analyze changes in data.

• There are advantages and disadvantages to using different mathematical tools to solve problems.

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Topic 4: Financial Literacy: Accounts, Credit, and CareersELPS: 1.A, 1.C, 1.D, 1.E, 1.G, 2.C, 2.D, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 4.A, 4.B, 4.C, 4.D, 4.G, 4.K, 5.E

1Knowledge You Can Bank OnChecking Accounts

Students are exposed to a checking account. They analyze the components of a check and write checks. They interact with a checkbook register by completing the balance column. The basics of reconciling a bank statement are introduced by comparing a portion of a checkbook register to a portion of a bank statement. The terms account balance, deposit, withdrawal, debit and transfer are defined. The concept of an overdraft is explained and an example is provided. Students then discover the costs and possible earnings involved in having a checking account. They compare checking accounts based upon the monthly fees, APY, and required minimum average balance to decide which account would be most appropriate.

• A checking account allows customers to safely store money in the bank and write checks against the money that they deposit.

• A statement is a monthly summary of the account balance on the checking account, including all transactions that occur during a given time period. It allows the customer to check their records against the bank’s records.

• A customer may have to pay money to have a checking account. The bank may charge a monthly fee just to have the account. Sometimes customers are required to keep a minimum amount of money in their account at all times. All banks charge a fee for overdrafts.

• A customer may earn money by having a checking account. Some banks offer an annual percentage yield (APY); this is a small percentage of interest based upon the customer’s account balance.

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2You Are a Real Card!Debit Cards vs. Credit Cards

Students compare and contrast the key characteristics of debit cards and credit cards. They practice methods for understanding provided characteristics, and they must determine if each characteristic applies to debit cards, credit cards, or both. They investigate credit cards in more depth as they deal with the financial advantages of rewards programs and the financial disadvantages of annual fees and interest rates.

Students use mathematics to see the increased cost of paying using credit cards over time. Students are introduced to the concepts of interest and interest rate. They investigate what happens if you pay only the minimum payment on your credit card balance. Students then determine the interest paid each month for given scenarios. Finally, they give advice to a customer who is looking to pay off his credit card.

The lesson ends with students again completing an activity in which they choose a debit card, credit card, or both, but this time they are given situations rather than characteristics, and they must apply their knowledge of the different types of cards. Then, students have a discussion of the advantages and disadvantages of both debit and credit cards.

• Debit cards are issued by the bank when a customer opens a checking account. When a customer buys an item using a debit card, the money is taken directly from their checking account. The amount of money the customer can spend is limited to the balance in their checking account.

• Credit cards are issued by a company when a customer applies to a financial/credit card company. When a customer buys an item using a credit card, they can make the purchase whether they currently have the money for it or not. The credit card company then bills the customer allowing them to pay over time. The amount of money the customer can spend is based upon the limit the credit card company provides.

• There are advantages to having a credit card. A customer can make a purchase without having the money available. Credit card companies offer rewards programs for using their cards. Also, for security purposes, the customer must sign each time they use the card. If the card is lost or stolen and used fraudulently, the customer is not responsible to pay for that illegal use of the card.

• There are disadvantages to having a credit card. A customer might make purchases they cannot afford. The credit card companies may charge an annual fee to have the card, and all credit card companies charge interest on any bill not paid in full at the end of each month.

• There are advantages to having a debit card. A customer can make a purchase just as if they were paying with cash, but they do not have to carry cash with them. Also, because the customer must have the funds available to make the purchase, there are limits to their ability to overspend.

• There is a disadvantage to using a debit card. For security purposes, the customer must enter a personal identification number (PIN) each time they use the card. However, if the card is lost or stolen and used fraudulently, the customer may not be able to recover the funds lost.

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3Financial Report CardUnderstanding Credit Reports

Students learn what a credit report is and why the credit score listed in the report is important. The credit score determines: (1) if the person qualifies for a loan, (2) the amount of money they can borrow, and (3) the interest rate for the loan. Students research what is included in a credit report. They discuss the range of credit scores, the importance of a good credit score, and how to maintain a good credit score.

Students interact with a circle graph to see what factors have the most impact on a credit score. They estimate the percent of each factor from the circle graph, rather than solving for the percents mathematically. They apply this information by ranking statements about a person’s credit report, connecting the data from the statements to a factor that affects a credit score, and then ranking the statements according to their importance in determining a credit score.

The lesson concludes with a brief discussion of how to earn a good credit score and the importance of having a positive credit report.

• A credit report is a detailed listing of an individual’s credit history, along with a credit score.

• A credit score is a number used by lenders to rate how likely a person is to repay their debts. The credit score determines: (1) if the person qualifies for a loan, (2) the amount of money they can borrow, and (3) the interest rate for the loan.

• It is important for individuals to have and maintain a good credit score, so that they can qualify for loans for a reasonable amount of money at a low interest rate.

• A good credit score is obtained by paying bills on time and avoiding having too much debt.

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4The Possibilities Are EndlessCareer Exploration

This lesson addresses career choice primarily from an educational and financial perspective. The students begin by completing a survey reflecting The first activity in the lesson helps students come to the understanding that the more education or training a person receives, the greater their potential earning power. Different post-secondary education degrees are defined, and a table comparing the financial benefits of having each of the degrees is provided. Students apply the information in the table and their knowledge of percents to compare incomes of jobs requiring different levels of education.

Students further investigate the finances of a career choice by taking into account both the potential earning power of having an increased level of education, and the cost of student loans to acquire that level of education. They are given personal scenarios and calculate the individual’s total earnings over a period of years.

• The more education or training that you receive, the greater your potential earning power.

• When considering the finances of a career choice, one must take into account both potential earning power of having an increased level of education, and the cost to acquire that level of education.

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5Student Aid 101Paying for College

Tuition is defined, and the methods to fund tuition, such as personal savings, grants, scholarships, and work-study programs are explained. Students then research different scholarships and how to locate grant money.

Students are given personal scenarios that include the cost of tuition, financial aid, scholarships, work-study opportunities, and grants. They use mathematics to determine the amount of the given financial aid packages, as well as how each package offsets the cost of tuition, in order to make financial decisions.

Next, the difference in tuition costs for a private school and a public school are explained, as well as the financial benefit for attending a school in-state. Students use mathematics to make comparisons of college tuition based upon in-state and out-of-state rates.

The lesson concludes with students putting together all that they have learned in this topic. Each student chooses the career(s) that interests them. Students use their chosen career(s) to: (1) describe the post-secondary education required, (2) estimate the cost of tuition, (3) determine a possible average yearly salary, (4) estimate the lifetime income, and (5) determine the average starting salary. Students also address in a general sense how to finance a college education.

• Tuition cost is one of the many factors to consider when selecting an appropriate post-secondary school.

• Students pay for college tuition in a variety of ways, including personal savings, grants, scholarships, work-study programs, and/or student loans.

• Once a student is accepted into a post-secondary institution, the institution proposes a financial aid package. The financial aid package includes a combination of scholarships, grants, and/or work-study programs that the student can use to offset the cost of tuition.

• The financial aid package may not cover the entire cost of tuition, so the student is responsible to pay the remaining portion of the tuition through savings, student loans, and funds from a part-time job. The student can also pursue additional scholarship opportunities on their own.

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5 Describing Variability of QuantitiesTopic 1: The Statistical ProcessELPS: 1.A, 1.B, 1.C, 1.D, 1.E, 1.G, 2.A, 2.C, 2.D, 2.E, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 3.G, 3.J, 4.A, 4.B, 4.C, 4.D, 4.G, 4.K, 5.A, 5.B, 5.E


1What's Your Question?Understanding the Statistical Process

Students consider a variety of questions and determine which are statistical and which are not. They learn about the statistical process: formulating a question, collecting data, analyzing data, and interpreting the results. Students organize data into two types: categorical and quantitative. Then, they determine whether conducting a survey, performing an experiment, or using an observational study would be the best method to answer each question. Students then conduct a survey in their classroom, interpret bar graphs and circle graphs for categorical data, and create a bar graph or circle graph for their survey data. Given a frequency table for categorical data, students determine relative frequency. Finally, students interpret the results, stating conclusions they can make from their data displays. In subsequent lessons, students will interpret histograms and dot plots for discrete quantitative data and histograms, stem-and-leaf plots, and box plots for continuous quantitative data.

• The statistical process includes formulating a statistical question, collecting data, analyzing the collected data, and interpreting the data in context of the situation.

• A statistical question is one that anticipates and accounts for variability in data.

• Data can be described as being categorical or quantitative. Categorical data is a set of data for which each piece of data fits into exactly one of several different groups or categories. Quantitative data is a set for which each piece of data can be placed on a numerical scale.

• Data are collected through the use of surveys, observational studies, and experiments.

• Categorical data can be displayed in tables, bar graphs, and circle graphs.

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2Get in ShapeAnalyzing Numerical Data Displays

Students examine data organized in a table and interpret a graphical representation of the same data in a dot plot. They interpret another table of data and then construct a dot plot for a given data set. Students then discuss the shape of data displayed in a dot plot and identify other properties such as clusters and gaps in graphs of data. Students examine a stem-and-leaf plot as a graphical representation of a larger data set. They interpret a table of data and create a stem-and-leaf plot for a given data set.

• Dot plots are a type of graph used to represent the frequency of data values using a number line.

• Dot plots are used to represent quantitative data, rather than categorical data. They are best suited for a small number of data points.

• Data sets have a graphical distribution, which can be described in terms of overall shape and pattern, as well as deviations from the pattern.

• Distributions are commonly referred to as symmetric, skewed left, skewed right, or uniform.

• Common graphical features include clusters, peaks, gaps, and outliers in the data values. Often a gap in the data is an indicator that the data include an outlier.

• Stem-and-leaf plots are a type of graph used to represent and organize data values for a large number of quantitative data.

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3

Follow Me on HistogramUsing Histograms to Display Data

Students analyze a histogram. They discuss intervals and interpret information from the histogram. Students then convert information from the histogram to a frequency table and compare the two representations. The process is reversed, and students create two histograms beginning with two tables of information. For each table, they convert the information to a frequency table, and finally, to a histogram. At the end of each problem, students summarize the data from the data displays.

• Histograms and bar graphs look very similar. Bar graphs are necessary when the data is categorical.

• Histograms are used when the data is numerical; numerical data can be represented continuously in intervals.

• The intervals in a histogram must all be the same size. The width of the bar represents the interval. The height of the bar indicates the frequency of values in the interval.

• Histograms and frequency tables display the same information. The histogram is a more visual representation of the information.

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Topic 2: Numerical Summaries of DataELPS: 1.A, 1.B, 1.D, 1.E, 1.G, 1.H, 2.A, 2.B, 2.C, 2.D, 2.F, 2.G, 2.H, 2.I, 3.A, 3.B, 3.C, 3.D, 3.F, 3.G, 3.J, 4.A, 4.B, 4.C, 4.E, 4.G, 4.J, 4.K, 5.A, 5.B, 5.E


1In the MiddleAnalyzing Data Using Measures of Center

Students analyze and interpret data using the three different measures of center: mode, median, and mean. Mode was defined in the previous module, but is now placed in the context of being one of three measures of central tendency. Median is defined in the first activity. Students calculate each numerical value and interpret its meaning in terms of a problem situation. Next, students investigate the third measure of center, mean, in three different ways: by leveling off a set of data values (creating fair shares), establishing a balance point for a set of data values, and determining the sum of the data values and dividing by the number of data values. In the last activity, students are presented with a real-world problem in which they calculate and interpret each measure of center.

• A measure of center for a data set summarizes all of its values with a single number.

• Measures of center are numerical ways of determining where the center of data is located.

• Three measures of center are mode, median, and mean.

• The mode is the data value or values that occur most often in a data set.

• The median is the data value in the middle of a data set that has been placed in numerical order.

• The mean can be thought of as leveling off a set of data values, a balance point of a data set placed on a number line, and the sum of the data values divided by the number of data values.

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2Box It UpDisplaying the Five-Number Summary

Students examine variability in data. They compute the range for a set of data and practice dividing data sets into quartiles. Students label and interpret the quartiles, and identify and interpret data that has been divided into different quartiles, called the five-number summary.

Students calculate the five-number summary and construct its accompanying box plot. They then construct and interpret box plots for a real-world situation.

Finally, students analyze two box plots, construct a plot given specific information, and use a box plot to create possible data sets.

• Measures of variability in a data set describe how spread out the data is.

• Quartiles are values that divide a data set into four parts once the data is arranged in ascending order.

• The five-number summary for a data set consists of the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value of the data set.

• The interquartile range, or IQR, is the the difference between the first and third quartiles (Q3 2 Q1).

• A box plot is another way to display numerical data on a number line. It displays the five-number summary and the interquartile range.

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3Dealing with DataCollecting, Displaying, and Analyzing Data

Students organize data into two types, categorical and quantitative. Quantitative data is further divided into discrete, continuous, and two-variable data. Students interpret graphs that represent all data types. They interpret bar graphs and percent bar graphs for categorical data.

• Data can be described as being categorical or quantitative. Categorical data is a set of data for which each piece of data fits into exactly one of several different groups or categories. Quantitative data is a set for which each piece of data can be placed on a numerical scale.

• Categorical data can be displayed in tables, bar graphs, and percent bar graphs.

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End of Course TopicFormative AssessmentELPS: 1.A, 1.C, 1.D, 1.F, 1.G, 1.H, 2.A, 2.B, 2.C, 2.D, 2.E, 2.G, 2.H, 3.B, 3.C, 3.D, 3.E, 3.F, 3.G, 3.H, 3.I, 3.J, 4.D, 4.E, 4.H, 4.I, 4.K, 5.B, 5.E, 5.F, 5.G


1Living Room RugPerformance Task

Students are provided with a scenario about Charlotte, who is looking to purchase a new rug for her living room. Charlotte wants her rug to have the same width-to-length ratio as her floor. Students are provided with the dimensions of the floor, along with a table of potential rug choices. They apply their knowledge of equivalent ratios to determine which rug(s) are proportional to Charlotte’s floor. Students support their reasoning by modeling the ratio relationships with both a double number line and a coordinate grid. Finally, students compare the cost for two rugs to determine which rug Charlotte should purchase.

• A ratio is a comparison of two quantities that can represent part-to-whole or part-to-part relationships.

• Ratios are used to represent proportional relationships in the real world.

• A proportion is an equation that states two ratios are equal.

• When writing a proportion, the numbers representing the same quantity must be placed in both numerators or in both denominators. The unit of measurement must be consistent among the ratios.

• Scaling up means to multiply both parts of a ratio by the same factor greater than 1, or divide both parts of a ratio by the same factor less than 1.

• Scaling down means to divide both parts of a ratio by the same factor greater than 1, or multiply both parts of a ratio by the same factor less than 1.

• A double number line is a model that is made up of two number lines used to represent the equivalence of two related numbers. The intervals on each number line maintain the same ratio.

• Equivalent ratios can be represented by tables, double number lines, and on coordinate planes.

• A ratio y—x is plotted as the ordered pair (x, y). • Equivalent ratios represented on the coordinate

plane form a straight line that passes through the origin.

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2Pizza PartyPerformance Task

Students are provided with a scenario about camp counselors who are planning a pizza party. The counselors survey everyone who will attend to find out which type of pizza they would prefer. Some of the data collected are displayed as the number of votes, and some of the data are displayed as percent of votes. Students apply their knowledge of part-whole relationships and percents to determine missing table values. Students then determine the number of people who changed their vote when given a new percentage to represent one part, or type of pizza. Finally, students apply their knowledge of proportions to determine the new whole, or total number of people attending, when provided with additional information.

• Percent is a fraction that represents a part-to-whole ratio with a whole of 100. The symbol % means “out of 100.”

• Percents can be expressed as fractions or decimals.• Percent problems involve three quantities: the part,

the whole, and the percent. • When calculating the part, given the percent and

the whole, write the percent as a decimal (move the decimal point two places to the left or divide by 100) and multiply by the whole.

• When calculating the percent, given the part and the whole, divide the part by the whole. Then move the decimal point two places to the right, or multiply by 100.

• When calculating the whole, given the percent and the part, write the percent as a fraction with a denominator of 100 and set it equal to the part over x. Then solve for x.

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3School Ski TripPerformance Task

Students are provided with a scenario about an upcoming ski trip at Kaitlynn’s school, which is organized by the PTA. Students are given information about the number of chaperones, number of students per chaperone, total cost, number of students per hotel room, number of passengers per bus, and costs associated with snow tubing. Students define variables, write equations, and solve for the unknown quantity in a series of situations related to the ski trip. For each equation, students interpret their solution and explain their reasoning.

• Equations can be used to represent and solve real-world problems.

• Variables can be used to represent quantities in expressions describing real-world values.





• The mathematical solution of an equation may not always be the same as the solution in a real-world context.

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topic 1: factors and multiples

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